Some Properties of the Capacity Value Function

Abstract

In a previous paper [5], two of the authors developed a method for ascribing a value to an extra unit of capacity on a telecommunications link. Specifically, they expressed the value of an extra unit of capacity as a function of current capacity, current occupancy and a planning horizon. The intention was to use this function as an ingredient in a bandwidth reallocation scheme for ensuring efficient operation of a telecommunications network. Unfortunately, direct evaluation of the function requires numerical inversion of a Laplace transform expressed in terms of Charlier polynomials, a task that is beyond the processing capabilities of typical switches in today's telecommunications networks. Because of this, it is desirable to have more easily computable methods of either calculating or approximating the capacity value function. We develop two approaches to this problem: the first is a recursive method of computing the Laplace transform of the capacity value function and the second is a linear approximation to the capacity value function itself.

Transient provisioning and performance evaluation for cloud computing platforms: A capacity value approach

“…The function g m,r x,y is reminiscent of the capacity value function of [12] and [13], and so we will also refer to it by that name. For a system with unit arrivals, no bursts in the arrival process, no buffer, no abandonments, and no capacity or buffer costs g m,r…”

Modelling complex stochastic systems: approaches to management and stability

“…Our work extends results in [115] and [116]. In [115] a Laplace transform expression in terms of Charlier polynomials is given for the undiscounted value of an additional unit of capacity during the planning horizon [0, t] on a link which never fails.…”

“…Due to PASTA there is a clear relationship between a system's stationary distribution and performance, however the relationship between the distribution of the number of jobs in the system at some finite time t and system performance is not as clear. In [115] and [116] a highly practical performance measure is introduced: the number of jobs rejected The analysis in [115] and [116] of a single resource loss system is based on the Laplace-Stieltjes transform, which for some function r : In Chapter 6 we generalise the results in [115] and [116] to the case of a single resource system where the resource experiences random periods of inaccessibility. These results are useful for incorporating faulty system behaviour into loss models.…”

“…The authors also demonstrate how the expression can be used to ascribe a value to an extra unit of capacity on the link over the planning horizon. In [116] a second order approximation to the inverted expression is given. The key difference between these papers and other classic studies of loss systems (see e.g., [109]) is that they focus on transient expected performance indicators rather 6.…”

“…Our methodological contribution is two-fold. First, we extend the model and results of [115] and [116] to allow for link failure. Second, we obtain an expression for infinite horizon total discounted lost revenue when interest is compounded continuously at rate r. As opposed to the expression for finite horizon undiscounted lost revenue, this expression does not require numerical inversion.…”